Causes of effects via a Bayesian model selection procedure

In causal inference, and specifically in the extit{Causes of Effects} problem, one is interested in how to use statistical evidence to understand causation in an individual case, and in particular how to assess the so-called {em Probability of Causation} (PC). The answer involves the use of potential responses, which describe what would have happened to the outcome if we had observed a different value for the exposure. However, even given the best possible statistical evidence for the association between exposure and outcome, we can typically only provide bounds for the PC. Dawid et al. (2016) highlighted some fundamental conditions, namely, exogeneity, comparability, and sufficiency, required to obtain such bounds from experimental data. The aim of the present paper is to provide methods to find, in specific cases, the best subsample of the reference dataset to satisfy these requirements. To this end, we introduce a new variable, expressing the preference whether or not be exposed, and we set the question up as a model selection problem. The best model is selected using the marginal probability of the responses and a suitable prior over the model space. An application in the educational field is presented.